2009 AMC 12A Problem 16

Question

A circle with center

C

is tangent to the positive

x

- and

y

-axes and externally tangent to the circle centered at

(3, 0)

with radius

1.

What is the sum of all possible radii of the circle with center

C?

3

4

6

8

9

A circle tangent to both positive axes with radius

r

has center

(r, r)

External tangency gives distance

r + 1

between

(r, r)

and

(3, 0);

form a quadratic in

r

and use Vieta

A circle tangent to both positive axes with radius

r

has center

(r, r).

External tangency to the circle at

(3, 0)

of radius

1

means the distance between centers is

r + 1

:

(r - 3)^2 + r^2 = (r + 1)^2.

Expanding gives

r^2 - 8r + 8 = 0.

Both roots

r = 4 \pm 2\sqrt{2}

are positive, and by Vieta’s formulas their sum is

8.

Thus, the correct answer is D.

Problem 16 in other years